, Volume 164, Issue 3, pp 451–460 | Cite as

The many faces of interpolation

  • Johan van BenthemEmail author


We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.


Interpolation Definability Cross-model relations First-order fragments Meta-theory 


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  1. Alechina, N., & Gurevich, Y. (1997). Syntax vs. semantics on finite structures. Structures in Logic and Computer Science, 1997, 14–33.Google Scholar
  2. Andréka H., Németi I. and van Benthem J. (1998). Modal logics and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3): 217–274 CrossRefGoogle Scholar
  3. Barwise J. (1975). Admissible sets and structures. Springer, New York Google Scholar
  4. Barwise, J., Feferman, S. (Eds.). (1985). Model-theoretic logics. New York: Springer. Google Scholar
  5. Barwise J. and Perry J. (1983). Situations and attitudes. Bradford Books/The MIT Press, Cambridge, Mass Google Scholar
  6. Barwise J. and van Benthem J. (1999). Interpolation, preservation pebble games. Journal of Symbolic Logic 64: 881–903 CrossRefGoogle Scholar
  7. Beth E.W. (1953). On Padoa’s method in the theory of definition. Indagationes Mathematicae 15: 330–339 Google Scholar
  8. Bolzano, B. (1837). Wissenschaftslehre, Seidelsche Buchhandlung, Sulzbach. English translation, 1973. In J. Berg & B. Terrell (Eds.), Theory of science. Dordrecht: Reidel.Google Scholar
  9. Bradfield J. and Stirling C. (2006). Modal μ-calculi. In: Blackburn, P. and Wolter, F. (eds) Handbook of modal logic, pp 721–756. Elsevier, Amsterdam Google Scholar
  10. Craig W. (1957). Linear reasoning. A new form of the Herbrand-Gentzen theorem. Journal of Symbolic Logic 22(3): 250–268 CrossRefGoogle Scholar
  11. d’Agostino, G. (1998). Modal logic and non-well founded set theory: Bisimulation, translation, and interpolation. Dissertation, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  12. d’Agostino G. and Hollenberg M. (2000). Logical questions concerning the μ-calculus. Journal of Symbolic Logic 65: 310–332 CrossRefGoogle Scholar
  13. d’Agostino G. and Lenzi G. (2005). An axiomatization of bisimulation quantifiers via the μ-calculus. Theoretical Computer Science 338(1–3): 64–95 CrossRefGoogle Scholar
  14. Ebbinghaus H.-D. and Flum J. (1995). Finite model theory. Springer, Berlin Google Scholar
  15. Fagin R. (1976). Probabilities on finite models. Journal of Symbolic Logic 41: 50–58 CrossRefGoogle Scholar
  16. Feferman S. and Kreisel G. (1968). Persistent and invariant formulas for outer extensions. Compositio Mathematicae 20: 29–52 Google Scholar
  17. Glebskii Y.V., Kogan D.I., Liogonkii M.I. and Talanov V.A. (1969). Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics 5: 142–154 CrossRefGoogle Scholar
  18. Hendriks, L. (1996).Computations in propositional logic. Dissertation DS-1996-01, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  19. Hollenberg, M. (1998).Logic and bisimulation. Ph.D. thesis, University of Utrecht.Google Scholar
  20. Hoogland, E., Marx, M., & Otto, M. (1999). Beth definability for the guarded fragment. In Proceedings of LPA R’99 (pp. 273–285). Berlin: Springer.Google Scholar
  21. Lindström P. (1966). First-order predicate logic with generalized quantifiers. Theoria 32: 165–171 Google Scholar
  22. Lindström P. (1969). On extensions of elementary logic. Theoria 35: 1–11 CrossRefGoogle Scholar
  23. Mason I. (1985). Undecidability of the meta-theory of the propositional calculus. Journal of Symbolic Logic 50: 451–457 CrossRefGoogle Scholar
  24. Pitts A. (1992). On an interpretation of second order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57: 33–52 CrossRefGoogle Scholar
  25. Schwichtenberg H. and Troelstra A. (1996). Basic proof theory. Cambridge University Press, Cambridge, UK Google Scholar
  26. ten Cate, B. (2005).Model theory for extended modal languages. Ph.D. thesis, ILLC, Amsterdam.Google Scholar
  27. Benthem J. (1985). The variety of logical consequence, according to Bolzano. Studia Logica 44(4): 389–403 CrossRefGoogle Scholar
  28. van Benthem, J. (1997). Dynamic bits and pieces. Report LP-97-01, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  29. van Benthem, J. (2003). Is there still logic in Bolzano’s key? In E. Morscher (Ed.), Bernard Bolzano’s Leistungen in Logik, Mathematik und Physik (pp. 11–34). Sankt Augustin: Academia Verlag.Google Scholar
  30. Benthem J. (2005). Minimal predicates, fixed-points and definability. Journal of Symbolic Logic 70(3): 696–712 CrossRefGoogle Scholar
  31. Benthem J. (2007). Inference in action. Publications de l’Institut Mathématique 82(96): 3–16 CrossRefGoogle Scholar
  32. van Benthem, J., ten Cate, B., & Väänänen, J. (2007). Lindström theorems for fragments of first-order logic. In Proceedings LICS 2007.Google Scholar
  33. Visser, A. (1996). Uniform interpolation and layered bisimulation. In Gödel ’96 Brno (pp. 139–164). Berlin: Springer.Google Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute for Logic, Language and Computation (ILLC)University of AmsterdamAmsterdamThe Netherlands

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